( 60) 



case that the set is uniformly continuous, i. e. that for any quantity 

 f, however small it may be, a quantity a can be pointed out so 

 that in any interval of the size of <> not one of the functions of the 

 set has oscillations larger than e. Concerning infinite uniformly con- 

 tinuous sets of functions there is a theorem that if they are limited 

 (i.e., if a maximum value and a minimum value can be given 

 between which all functions move) they possess at least one continuous 

 limiting curve, to which uniform convergence takes place 1 ). 



We shall prove, that for the set of functions of the difference 

 quotients of a finite continuous function, if it be uniformly continuous, 

 follows in the first place the limitedness and furtheron for indefinite 

 decrease of the ^--increase the existence of only one limiting curve, 

 so that holds : 



Theorem 1. If a finite continuous function f(x) has a uniformly 

 continuous set of difference quotients, then it possesses a finite con- 

 tinuous differential quotient 2 ). 



To prove this we call q „j? a (x) the size of the region of oscillation 



between x and x -j- A of the difference quotient for an .x'-increase 6. 



If we allow 6 to assume successively all positive values, then it 



follows from the supposed uniform continuity, that A can always 



be chosen so small as to keep all values $ s A (x) below a certain 



amount as small as one cares to make it. If we thus call éa (x) the 



maximum of the values je (#) for definite x and A, then e \ (x) tends 



with A uniformly to 0. 



P 

 We have fartheron if — is a proper fraction : 



n 



<p A ( X ) = -1^ (X) + — ?>A 



n 



<p p ^ (X) = — tf± {X) + — (f A_( X 



If we break up each of the n terms of the second member of 

 (1) into p equal parts and each of the p terms of the second member 

 of (2) into n equal parts, then the difference of those two second 

 members can he divided into pn terms, each remaining in absolute 



value smaller than — . b (x), so that the difference of <p {x) and y p (x) 



pn ^ — A 



remains smaller than e A (x) in absolute value. 



!) Compare Arzela, u Sulle funzioni di limë\ Memorie della Accademia di Bologna, 

 serie 5, V, page 225. 



2 ) We suppose the function to be given in a certain interval of values of the 

 independent variable x. 



